Question 1

Given the function $$y = -5g(x)$$, how does the graph of $$y = -5g(x)$$ differ from the graph of $$y = g(x)$$?

Accepted Answer

The graph is reflected across the $$x-$$axis and stretched vertically by a factor of $$5.$$

Question 2

Which graph best represents the piecewise function $$u(x) = \begin{cases} 2x-5, & x \leq 0 \ -x-5, & x \u003e 0 \end{cases}$$?

Accepted Answer

The first image (in the set above).

Question 3

Given the ellipse $$4x^2$$ + $$y^2 = 4$, what is the formula and domain for g_3$(x) as $$shown in the fifth image?

Accepted Answer

$$g_3(x) = -2$$\sqrt{$$1-x^2$$}$$, Domain: -1$$ \leq x \leq 0$$

Question 4

Determine the type of symmetry for the graph represented by the equation $$4y^2$$ - $$x^2 = 16$.

Accepted Answer

Symmetric about the $$y-$$axis only

Question 5

Find the amplitude and period for the trigonometric function $$P(x) = 5.7\sin\left(\frac{1}{9}(x-7)\right).$$

Accepted Answer

Amplitude = $$5.7$$, Period = $$18\pi$$

Question 6

Let $$g(x) = \begin{cases} \frac{x^3$-1}{x-1}, & x 
eq 1 \ a, & x = 1 \end{cases}. $$For what value of $$a is$$ $$g(x)$$ continuous at $$x=1$$?

Accepted Answer

$$a = 3$$

Question 7

Evaluate $$\lim_{x 	o 0} \frac{\sin 39x}{9x}.$$

Accepted Answer

$$\frac{13}{3}$$

Question 8

Refer to the sixth image above. What is $$\lim_{x 	o 5} f(x)$$?

Accepted Answer

$$7$$

Question 9

Determine the derivative of $$f(x) = 4x^4$ using the definition of the derivative.

Accepted Answer

$$f'(x) = 16x^3$

Question 10

A basketball is shot upwards from the ground with an initial velocity of $$40 ft/s. $$The height (in feet) after $$t$$ seconds is $$h(t) = -16t^2$ + 40t. $$What is the maximum height reached by the basketball?

Accepted Answer

$$25 ft$$

Question 11

Determine the derivative of $$g(x) = \frac{4x^4$ + 3x^3$ - 2x^2$}{x^4$}.$$

Accepted Answer

$$g'(x) = \frac{3}{x^2$} - \frac{4}{x^3$}$$

Question 12

Find the equation of the normal line to the curve $$x^4 - x$$ $$y^2$$ + $$y^4 = 24$ at the point (2$$, -2)$$.

Accepted Answer

$$y = -\frac{6}{7}x + \frac{16}{7}$$

Question 13

Determine the equation of the tangent line to $$y = x\sqrt{17-x^2$} at $$the point $$(1, 4).$$

Accepted Answer

$$y = \frac{15}{4}x - \frac{1}{4}$$

Question 14

Use linear approximations to estimate the value of $$e^{-0.05}. $$Choose the value of $$a$$ that minimizes the error.

Accepted Answer

$$a = 0.95$$

Question 15

A cylindrical swimming pool with a radius of $$2$$ meters and a depth of $$5$$ meters is being emptied. If the water level is dropping at a rate of $$3 cm/$$min, at what rate is the water leaving the pool in cubic meters per minute?

Accepted Answer

$$\frac{-3\pi}{25} m$^3/$$min

Question 16

Provide the differential expression $$dy$$ for the function $$f(x) = 4\ln(2-3x). $$

Accepted Answer

$$dy = \frac{4}{2-3x} dx$$

Question 17

Find the second derivative $$y''$$ for the function $$y = 25\cot x.$$

Accepted Answer

$$y'' = -25\csc^2$ x \cot x$$

Question 18

Consider the function $$f(x) = \cos x on $$the interval $$[0, 2\pi]. $$Using the difference quotient $$D(x) = \frac{f(x+0.02) - f(x)}{0.02}$$, what does the graph of $$D(x)$$ resemble?

Accepted Answer

The graph of $$-\sin x$$

Question 19

Determine the equation of the line perpendicular to the tangent line of the curve $$y = 2x + 5 at $$the point $$Q(2, 9).$$

Accepted Answer

$$y = \frac{1}{2}x + 8$$

Piecewise Functions: Sketching and Interpretation

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